Question: All of the 5th grade teachers and students from Almond went on a field trip to an archaeology museum. Tickets were $$7.50$ each for teachers and $$2.00$ each for students, and the group paid $$31.00$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$15.00$ each for teachers and $$8.50$ each for students, and the group paid $$98.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7.5x+2y = 31}$ ${15x+8.5y = 98}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-15x-4y = -62}$ ${15x+8.5y = 98}$ Add the top and bottom equations together. $ 4.5y = 36 $ $ y = \dfrac{36}{4.5}$ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into $ {7.5x+2y = 31}$ to find $x$ ${7.5x + 2}{(8)}{= 31}$ $7.5x+16 = 31$ $7.5x = 15$ $x = \dfrac{15}{7.5}$ ${x = 2}$ You can also plug ${y = 8}$ into $ {15x+8.5y = 98}$ and get the same answer for $x$ ${15x + 8.5}{(8)}{= 98}$ ${x = 2}$ There were $2$ teachers and $8$ students on the field trips.